361 reputation
312
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location United Kingdom
age 43
visits member for 4 years, 4 months
seen Nov 22 at 1:19

Jul
21
answered Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?
Jul
2
awarded  Curious
Mar
4
comment Are there more rational numbers than integers?
@caters - possibly simpler way to put it - there's at least one way to biject rationals to/from positive integers. It doesn't matter that you can't determine which rational has count n without first checking all the rationals for counts 1..n-1 for possible duplicates. Having a non-recursive way to compute the mapping isn't a requirement.
Mar
4
comment Are there more rational numbers than integers?
@caters - usual counting system for positive rationals - 0/1, 1/1, 2/1, 1/2, 3/1, 1/3, 4/1, 3/2, 2/3, 1/4, ... - all the rationals with numerator+denominator=1, then those with numerator+denominator=2, then those with numerator+denominator=3 and so on, but excluding those that are equivalent to rationals already counted. You can count as far as you want, though you can never finish counting - just like counting positive integers. To include negative rationals, just alternate positive then negative in the counting sequence.
Mar
4
comment Are there more rational numbers than integers?
@caters - I think you're confusing "countable" with "finite". Countable means you can think of a scheme for counting that theoretically counts all of them - it doesn't mean the process of counting them has to be achievable. As for doing operations with rationals - actually, rationals are closed over all arithmetic operations, excluding division by zero of course. If start with a counted sequence of rationals 1..n, operations may result in rationals with a count larger than n, but that's just it - rationals are a countably infinite set. There's no requirement for any finite subset to be closed.
Oct
22
awarded  Nice Question
Sep
27
accepted Is there a name for functions that are their own inverses
Sep
27
comment Is there a name for functions that are their own inverses
Thanks - have to wait a few minutes before I can accept, but this is the answer.
Sep
27
asked Is there a name for functions that are their own inverses
Aug
26
awarded  Notable Question
Jun
4
awarded  Teacher
May
26
accepted Which polygon tile grids allow convex polygons to be formed from multiple tiles?
May
23
comment Which polygon tile grids allow convex polygons to be formed from multiple tiles?
The two pathfinding ideas are separate anyway. Tthe whole point of defining convex cells is that there are no pre-defined cells, and the whole point of the two-step non-convex cells is to cope with e.g. circular barriers - if everything is defined on a square/whatever grid these should be non-issues. Basically, I was just going off on a random tangent. maybe thinking about optimising pathfinding on a grid by combining cells. Though this certainly shows a case where larger convex cells are possible, but can't be found by growing incrementally larger convex cells.
May
23
comment Which polygon tile grids allow convex polygons to be formed from multiple tiles?
Very nice, but no. The idea started when I was reading a blog post about pathfinding in square grids, and I wondered how far I could generalise to different grids. I thought about an idea I had some time ago about defining cells for the pathfinding to work on where each cell was as large as possible while being convex (so any point within a cell could be reached from any other in one step - I also had a variant where any point could be reached from any other in two steps via a centre point).
May
22
awarded  Citizen Patrol
May
22
answered Which polygon tile grids allow convex polygons to be formed from multiple tiles?
May
22
asked Which polygon tile grids allow convex polygons to be formed from multiple tiles?
Apr
11
comment Are there more rational numbers than integers?
Starting from integers, your extra degree of freedom can derive new values - but that's just saying that the set of rationals is a strict superset of the set of integers. That gives a sense in which the set of rationals is larger, but that sense is not the same as set cardinality - the two versions of size are equivalent for finite sets, but not (at least in general) for infinite sets. The strict-superset ordering isn't even fully defined, and as most mathematicians consider "size" and "cardinality" synonyms, using "size" for something else causes confusion and grumpiness.
Apr
11
comment Are there more rational numbers than integers?
I like the two dimensions/two degrees of freedom argument, but it's really the same thing I was thinking about when I revisited this. On the one hand, two dimensions/degrees of freedom needn't give more meaningful choice. Consider the ratio of two reals - you still have two degrees of freedom, yet the result is always a real (unless the denominator is zero). You don't get more values - the apparent extra choice is an illusion.
Mar
10
awarded  Notable Question